mod2opd

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mod2opd computes the Zernike function evaluation matrix Z described previously.  This matrix works together with the V matrix computed by slope2mod.  After slope2mod and mod2opd have been executed, the user can obtain the final reconstructed wavefront by performing the multiplication

      dopd, rec  =  Z V sdata ,  where dopd, rec  will be in m, if sdata is in rad.

From the Matlab command line, the calling syntax for mod2opd is:

>> z = mod2opd(xopd, yopd, orders, rad, arad);

where

xopd  = vector (dim nopd x 1) containing the x-coordinates of the points at which WF opd estimation is desired.  A simple way of generating xopd is to set it equal to the xact vector generated by aogeom, where the "actuator" points have been defined to be at the corners of the subapertures.  However, the Zernike expansion may be evaluated at any arbitrary {xopd,yopd} the user cares to specify.

yopd  = vector (dim nopd x 1) containing the y-coordinates of the points at which WF opd estimation is desired.  A simple way of generating yopd is to set it equal to the yact vector generated byaogeom, where the "actuator" points have been defined to be at the corners of the subapertures.  However, the Zernike expansion may be evaluated at any arbitrary {xopd,yopd} the user cares to specify.

orders = vector of indices [r1,r2,...,rN ], that specifies the radial order of the Zernike modes to be included.  For any included radial order, all azimuthal (angular) orders are included.

The values r1,r2,...,rN need not be consecutive: any radial order(s) may be omitted.  Radial order 1, referring to tilt, is the lowest allowed order.

rad   = radius (in m) of the circle that defines the normalizing radius for the Zernike circle functions.  This should match the radius over which the reconstruction is desired.

arad = radius (in m) of the inner obscuration, if annular aperture is desired.

          Omit this argument to use the basic Zernike set for unobstructed circular pupil.

z  = the matrix Z that contains the values of the Zernike basis functions at the specified opd points.

      The defining relation is Zi,k = Zk(ri),  where ri = (xopdi, yopdi),  i=1,...,nopd.

 

Zernike normalization and ordering convention

The optics literature contains numerous variations on the convention for normalizing and ordering the Zernike circle functions.  In the AO tools routines that deal with modal decomposition, we essentially follow the Zernike convention that is used by Born and Wolf, or by Malacara, but with a slight variation as used by Gavrielides.  We keep full consistency with Gavrielides' conventions since we use his method of projecting the Zernike coefficients directly from slope data.

Note that the normalization and ordering convention of the Zernikes is immaterial when computing the full reconstruction formula

      dopd, rec = Z amod, rec = Z V sdata

As long as Z and V are consistent, they produce the correct dopd, rec  in meters.  However, if one wants to work with the amod, rec  values themselves, then one must understand the ordering and normalization convention, in order to correctly interpret the values {a1, a2, ...}.

      The remainder of this section is under construction.

REFERENCES:

M. Born and E. Wolf, Principles of Optics, 5th ed., pp. 464-468, 1975.

D. Malacara and S.L. DeVore,  "Interferogram evaluation and wavefront fitting", in Optical Shop Testing, 2nd ed., ed. D. Malacara, pp. 461-472, Wiley, 1992.

V.N. Mahajan,  "Zernike annular polynomials for imaging systems with annular pupils", J. Opt. Soc. Am., vol. 71, no. 1, pp. 75-85, Jan. 1981.

V.N. Mahajan,  "Zernike annular polynomials and optical aberrations of systems with annular pupils", Appl. Opt., vol. 33, no. 34, pp. 8125-8127, Dec. 1994.